ACT Math Formulas

Quadratic Formula

For a quadratic equation ax² + bx + c = 0, the solutions for x can be found using the quadratic formula.

Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Example: Solve 2x² + 4x – 6 = 0. Here a = 2, b = 4, c = –6. Plugging into the formula: $x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 2 \cdot (-6)}}{2 \cdot 2} = \frac{-4 \pm \sqrt{16 + 48}}{4} = \frac{-4 \pm \sqrt{64}}{4} = \frac{-4 \pm 8}{4}$, so x = 1 or x = -3. The ACT often includes at least one quadratic equation to solve, so memorizing this formula is crucial.

(Also know basic algebra skills like factoring and exponent rules, even if they're not "plug-in" formulas.)

Slope of a Line

The slope formula gives the steepness of a line through two points. It's the change in y divided by the change in x (often remembered as "rise over run").

Formula: $\text{Slope } (m) = \frac{y_2 - y_1}{x_2 - x_1}$

Example: If you have two points (2, 5) and (6, 13), the slope is $\frac{13 - 5}{6 - 2} = \frac{8}{4} = 2$. A typical ACT question might give two points and ask for the slope of the line connecting them.

Distance Formula

This formula finds the distance between two points on the coordinate plane.

Formula: $\text{Distance } (d) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Example: For points (1, 2) and (5, 5), the distance is $\sqrt{(5-1)^2 + (5-2)^2} = \sqrt{16 + 9} = \sqrt{25} = 5$.

Midpoint Formula

This gives the midpoint (the point exactly in between) of two coordinates by averaging their x-values and y-values.

Formula: $\text{Midpoint} = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$

Example: For points (1, 2) and (5, 5), the midpoint is $(\frac{1+5}{2}, \frac{2+5}{2}) = (3, 3.5)$.

Area of a Triangle

The area of a triangle is half the base times the height.

Formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

Example: For a triangle with base 10 and height 6, area = $\frac{1}{2} \times 10 \times 6 = 30$.

Pythagorean Theorem

In a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse (longest side).

Formula: $a^2 + b^2 = c^2$

Example: If a right triangle has legs 5 and 12, the hypotenuse $c = \sqrt{5^2 + 12^2} = 13$.

Special Right Triangles

Memorize the common side ratios for these special triangles:

• 45°-45°-90°: sides x, x, x√2 (hypotenuse = x√2)
• 30°-60°-90°: sides x, x√3, 2x (hypotenuse = 2x)

(For example, a 30-60-90 triangle with short side 5 will have the other leg 5√3 and hypotenuse 10.)

Area of a Circle

The area inside a circle is π times the radius squared.

Formula: $\text{Area} = \pi r^2$

Example: A circle with radius 4 has area 16π.

Circumference of a Circle

The circumference (perimeter) of a circle is π times the diameter (or 2π times the radius).

Formula: $\text{Circumference} = 2\pi r$

Example: If a circle has radius 7, its circumference is 2π × 7 = 14π.

Sine, Cosine, Tangent (SOH-CAH-TOA)

In a right triangle, these define ratios of a given angle's sides:

• sin θ = opposite / hypotenuse
• cos θ = adjacent / hypotenuse
• tan θ = opposite / adjacent

Example: If a right triangle has an acute angle θ with opposite side 5 and hypotenuse 13, then sin θ = 5/13.

(It's also useful to remember that sin²θ + cos²θ = 1, in case you need to find one ratio from the other.)

1. Identify the Concept

Read the question carefully and determine what it's asking. Look for keywords that hint at a formula or concept. For example, words like "average," "slope," "distance," "area," or "probability" indicate which formula you might need. Training yourself to quickly recognize the type of problem will tell you which formula to use.

2. Write Down the Formula

Once you know which formula is relevant, jot it down on scratch paper (or visualize it) before plugging in numbers. This helps prevent mistakes and ensures you're applying the formula correctly under time pressure.

3. Plug in the Values

Substitute the given values into the formula and solve step-by-step. Work carefully to avoid arithmetic errors. For instance, if a problem gives two points and asks for the distance, write down the distance formula and plug in the coordinates. Laying out the steps can help catch mistakes and makes complex problems more manageable.

4. Use Reasoning if You Forget

If a formula slips your mind, try to derive it or use logic. For example, if you forget the trapezoid area formula, you can find the area by splitting the trapezoid into simpler shapes (rectangles and triangles) and adding them up. Likewise, if you forget a geometry formula, drawing a quick sketch and recalling basic principles (like the Pythagorean theorem) can guide you to the answer.

5. Manage Your Time

Don't spend too long on one problem. Each math question is roughly one minute. If applying a formula is taking too long or you're unsure which formula to use, skip the question and come back if time allows. Sometimes a later question might even jog your memory for a formula you skipped. Keep moving to ensure you get to every question.

Focus on High-Yield Formulas First

Prioritize memorizing formulas that are used frequently on the ACT. For example, formulas for linear equations, basic geometry, and trig appear often, whereas something like the equation of a circle might appear only once. Master the basics before worrying about very rare formulas.

Use a Few Each Day

Consistency is key. Spend some time each day studying a handful of formulas rather than cramming once a week. Even 5-10 minutes daily can be more effective. This steady practice helps move formulas into your long-term memory.

Use Flashcards or a Formula Chart

Test yourself by writing formulas on flashcards (formula name on one side, the formula on the other) or by repeatedly writing down a formula sheet from memory. Regular self-quizzing will reinforce your recall and highlight which formulas you need to work on.

Understand the Formula (Don't Just Memorize)

Take time to understand why each formula works. If you know that the distance formula comes from the Pythagorean theorem, it's easier to remember. Understanding connections helps you re-derive a formula if you forget it, and it reinforces your memory.

Practice with Real ACT Questions

Apply these formulas on actual ACT practice problems or past tests. This will reinforce when and how to use each formula under exam conditions. After practice tests, review any questions you missed and make sure it was not due to forgetting a formula. This helps you identify which formulas to review before test day.